Inflow Forecasting for Hydropower Operations: Bayesian Model Averaging for Postprocessing Hydrological Ensembles

Transcription

1 Inflow Forecasting for Hydropower Operations: Bayesian Model Averaging for Postprocessing Hydrological Ensembles Andreas Kleiven, Ingelin Steinsland Norwegian University of Science & Technology Dept. of Mathematical Sciences Dept. of Industrial Economics and Technology Management 12 September 2018

2 Introduction Inflow forecasting methods: Data-driven methods and physically-based methods Ensemble predictions are commonly used to assess the uncertainty of future inflow Ensemble predictions are often biased and have too small variance In this work we develop statistical postprocessing methods to obtain calibrated and sharp probabilistic forecasts 2

3 Bayesian Model Averaging (BMA) f (y x 1,..., x M ) = Y x m N (µ m, τ 2 ) µ m = α m + β m x m Y : Random quantity of interest M m=1 1 w m g(y x m ) x m, m = 1,..., M: Deterministic predictions f (y x 1,..., x M ): Predictive distribution, given M predictions g(y x m ): Pdf associated with with member m w m : Weight associated with member m α m and β m : Bias correction parameters for member m τ : Standard deviation of the ensemble member pdfs 3

4 BMA with exchangeable ensemble members f (y x 1,..., x M ) = 1 M Y x m N (µ m, τ 2 ) µ m = α + βx m M g(y x m ) m=1 Probability Density Observation BMA pred. pdf Mean of pred. pdf Raw ens. Bias corr. ens. Ens. pdfs Inflow 2 3 4

5 Applications and earlier work Temperature forecasting Raftery, Gneiting, Balabdaoui, and Polakowski (2005) Precipitation forecasting Sloughter, Raftery, Gneiting, and Fraley (2007) Wind speed forecasting Sloughter, Gneiting, and Raftery (2010), McLean Sloughter, Gneiting, and Raftery (2013) Inflow forecasting Duan, Ajami, Gao, and Sorooshian (2007), Vrugt and Robinson (2007), Todini (2008), Rings, Vrugt, Schoups, Huisman, and Vereecken (2012) 5

6 Bayesian Model Averaging using Varying Coefficient Regression (BMA-VCR) BMA f (y x 1,..., x M ) = 1 M Y x m N (µ m, τ 2 ) µ m = α + βx m M g(y x m ) m=1 BMA-VCR f (y x 1,..., x M ) = 1 M Y x m N (µ m, τ 2 ) M g(y x m ) m=1 µ m = (α + α t ) + (β + β t )x m α t = α t 1 + a t, a t N(0, δ 1 ) β t = β t 1 + b t, b t N(0, δ 1 ) 6

7 Evaluation of probabilistic forecasts: Calibration P(rank) Rank P(rank) Rank P(rank) Rank P(rank) Rank 7

8 Evaluation of probabilistic forecasts: Sharpness Continuous ranked probability score (CRPS) Comparable to absolute error for deterministic forecasts For a perfect deterministic forecast, the CRPS is zero Cumulative density Probability density Observation BMA pred. pdf Mean of pred. pdf Raw ens. Bias corr. ens. Ens. pdfs 0 1 Inflow

9 Results Relative Frequency PIT values Relative Frequency Rank of observation in ensemble Mean CRPS ii BMA VCR BMA Ensemble Inverse precision parameter δ 1 9

10 Conclusions and future work We have presented a statistical postprocessing method based on BMA Demonstrated good results based on real data for low lead times For higher lead times, consider alternative pdfs associated with the ensemble members, e.g. a combination of beta distributions and the empirical distribution of historical inflow values Extend the method to a higher-dimensional system 10

11 References Q. Duan, N. K. Ajami, X. Gao, and S. Sorooshian. Multi-model ensemble hydrologic prediction using bayesian model averaging. Advances in Water Resources, 30(5): , J. McLean Sloughter, T. Gneiting, and A. E. Raftery. Probabilistic wind vector forecasting using ensembles and bayesian model averaging. Monthly Weather Review, 141(6): , A. E. Raftery, T. Gneiting, F. Balabdaoui, and M. Polakowski. Using bayesian model averaging to calibrate forecast ensembles. Monthly weather review, 133(5): , J. Rings, J. A. Vrugt, G. Schoups, J. A. Huisman, and H. Vereecken. Bayesian model averaging using particle filtering and gaussian mixture modeling: Theory, concepts, and simulation experiments. Water Resources Research, 48(5), J. M. Sloughter, T. Gneiting, and A. E. Raftery. Probabilistic wind speed forecasting using ensembles and bayesian model averaging. Journal of the american statistical association, 105(489):25 35, J. M. L. Sloughter, A. E. Raftery, T. Gneiting, and C. Fraley. Probabilistic quantitative precipitation forecasting using bayesian model averaging. Monthly Weather Review, 135(9): , E. Todini. A model conditional processor to assess predictive uncertainty in flood forecasting. International Journal of River Basin Management, 6(2): ,